Dividing Polynomials - Practice Questions

Dividing polynomials is the process of dividing one polynomial by another to find a quotient and, sometimes, a remainder. It is similar to dividing ordinary numbers.

Solved Examples

Problem 1. Using synthetic division, find the quotient and remainder of \bold{\frac{x^2 + 3}{x - 4}} 

Solution:

Dividend = x² + 3 or, x² + 0x + 3

Divisor = x - 4

Applying synthetic division, we have:

\begin{array}{c|rrr}&1&0&3\\4&&4&16\\\hline\\&1&4&19\\\end{array}

The first two numbers of the last row represent the coefficients of the quotient and the third value is the remainder.

Thus, the quotient is x + 4 and the remainder is 19.

Problem 2. Solve \bold{\frac{4x^3+5x^2+5x+8}{4x+1}}using long division.

Solution:

Dividend = 4x³ + 5x² + 5x + 8

Divisor = 4x + 1

Using long division method, we have:

\begin{array}{r} x^2+x+1\phantom{)}   \\ 4x+1{\overline{\smash{\big)}\,4x^3+5x^2+5x+8\phantom{)}}}\\ \underline{4x^3~\phantom{}+x^2~~~~~~~~~~~\phantom{-b)}}\\ 4x^2+5x~~~~~~~\phantom{)}\\ \underline{~\phantom{()}4x^2+1x~~~~~~~~~}\\ 4x+8\phantom{)}\\ \underline{-~\phantom{()}(4x+1)}\\ 7\phantom{)}\\ \end{array}

Thus, the quotient and remainder are x² + x + 1 and 7 respectively.

Problem 3. Solve \bold{\frac{4x^3-3x^2+3x-1}{x-1}} using synthetic division.

Solution:

Dividend = 4x³ - 3x² + 3x - 1

Divisor = x - 1

Applying synthetic division, we have:

\begin{array}{c|rrr}&4&-3&3&-1\\1&&4&1&4\\\hline\\&4&1&4&3\\\end{array}

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 4x² + x + 4 and the remainder is 3.

Problem 4. Solve \bold{\left( 5{{x}^{3}}-6{{x}^{2}}+3x+11 \right)\div \left( x-2 \right)} using synthetic division.

Solution:

Dividend = 5x³ - 6x² + 3x + 11

Divisor = x - 2

Applying synthetic division, we have:

\begin{array}{c|rrr}&5&-6&3&11\\2&&10&8&22\\\hline\\&5&4&11&33\\\end{array}

The first three numbers of the last row represent the coefficients of the quotient and the fourth value is the remainder.

The quotient is 5x² + 4x + 11 and the remainder is 33.

Problem 5. Solve \bold{\left( 18{{x}^{4}}+9{{x}^{3}}+3{{x}^{2}} \right)\div \left( 3{{x}^{2}}+1 \right) } using long division.

Solution:

Dividend = 18x⁴ + 9x³ + 3x² + 0x + 0

Divisor = 3x² + 1

Using long division method, we have:

\begin{array}{r} 6x^2+3x-1\phantom{)}   \\ 3x^2+1{\overline{\smash{\big)}\,18x^4+9x^3+3x^2+0x+0\phantom{)}}}\\ \underline{18x^4~\phantom{}+0x^3+6x^2~~~~~~~~~\phantom{-b)}}\\ 9x^3-3x^2+0x+0\phantom{)}\\ \underline{~\phantom{()}9x^3+0x^2+3x~~~~~~~~~}\\ -3x^2-3x+0\phantom{)}\\ \underline{~\phantom{()}-3x^2+0x-1}\\ -3x+1\phantom{)}\\ \end{array}

Thus, the quotient and remainder are 6x² + 3x - 1 and -3x + 1 respectively.

Problem 6. Given the polynomial p(x) = x² + x +5 and g(x) = x +2. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

q(x) = x - 1 and r(x) = 7. 

Problem 7. Given the polynomial p(x) = x³ + x + 6x² + 4 and g(x) = x² + 1. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

So, here q(x) = x + 6, and r(x) = -2. 

Problem 8. Given the polynomial p(x) = x⁴ + x + 6x² + 4 and g(x) = x² + 1. Find the value of q(x) and r(x). 

Solution: 

Using the steps mentioned above. On dividing p(x) with g(x) we get, 

So, here q(x) = x² + 5, and r(x) = x -1. 

Problem 9. Given the polynomial x⁴ - 1. We know two of the roots are -1, 1. Find the other two roots if they exist. 

Solution: 

We know that the two roots are -1 and 1. 

So, x -1 and x + 1 are the factors of the given polynomial. Then, (x - 1)(x +1) is also a factor of the polynomial. 

(x - 1) (x + 1) = x² - 1

We see that quotient is x² + 1. The roots of this polynomial will be the roots of the equation. 

x² + 1 = 0

⇒ x² = -1 

This equation cannot have real roots, thus roots don't exist for this polynomial. 

Problem 10. Given the polynomial p(x) = x⁵ + 8x³ - 6x⁴ + 5x² + 10x + 8 and g(x) = x² + 10x -5. Find q(x) and r(x).

Solution: 

Using the same steps as done in previous questions, 

q(x) = x³ - 16x² + 173x - 1805 

r(x) = 18295x - 9017

Problem 11. For the same given the polynomial p(x) = x⁵ + 8x³ - 6x⁴ + 5x² + 10x + 8 and g(x) = x + 5. Find q(x) and r(x).

Solution: 

Here, q(x) = x⁴ - 11x³ + 63x² - 310x + 1560 

r(x) = -7792 

Problem 12. For the same given the polynomial p(x) = x⁵ - 6x⁴ + 5x² + 8 and g(x) = x + 2. Find q(x) and r(x).

Solution: 

q(x) = x⁴ - 8x³ + 16x² -27x + 54 

g(x) = -100

Practice Problems

Problem 1: Find the quotient and remainder when 12345 is divided by 97.

Problem 2: Given that when 3456 is divided by 89 the remainder is 56. Find the quotient.

Problem 3: Divide 6789 by 123 and express the result in terms of quotient and remainder.

Problem 4: If a divided by 25 gives a quotient of 37 and a remainder of 8 find the value of a.

Problem 5: Determine the quotient and remainder when 98765 is divided by 543.

Problem 6: Find the remainder when 1001 is divided by 7 and 11. Use the results to find the quotient for each case.

Problem 7: When 2024 is divided by 40 the remainder is r. Find r and express 2024 in terms of the quotient and remainder.

Problem 8: If 5678 is divided by 54 what is the remainder? Also, find the quotient.

Problem 9: Find the quotient and remainder when 4321 is divided by 89.

Problem 10: Given that the quotient of 8765 divided by 89 is 98 find the remainder.